Fractal analysis of extracellular matrix for observer-independent quantification of intestinal fibrosis in Crohn’s disease

Prevention of intestinal fibrosis remains an unresolved problem in the treatment of Crohn’s disease (CD), as specific antifibrotic therapies are not yet available. Appropriate analysis of fibrosis severity is essential for assessing the therapeutic efficacy of potential antifibrotic drugs. The aim of this study was to develop an observer-independent method to quantify intestinal fibrosis in surgical specimens from patients with CD using structural analysis of the extracellular matrix (ECM). We performed fractal analysis in fibrotic and control histological sections of patients with surgery for CD (n = 28). To specifically assess the structure of the collagen matrix, polarized light microscopy was used. A score to quantify collagen fiber alignment and the color of the polarized light was established. Fractal dimension as a measure for the structural complexity correlated significantly with the histological fibrosis score whereas lacunarity as a measure for the compactness of the ECM showed a negative correlation. Polarized light microscopy to visualize the collagen network underlined the structural changes in the ECM network in advanced fibrosis. In conclusion, observer-independent quantification of the structural complexity of the ECM by fractal analysis is a suitable method to quantify the degree of intestinal fibrosis in histological samples from patients with CD.


What is the box-counting dimension?
Let us consider a binary image by which we mean a planar domain Ω, e.g. a square, such that every point x ∈ Ω is assigned a value 0 (black) or 1 (white).A binary image is therefore nothing else as a function f : Ω → {0, 1} such that f(x) is the color code.
One way to define the dimension of the white region of an image is to use the concept of fractal dimension or box-counting dimension, see [30] for applications to biology.We cover the white region of an image by boxes of side length ε.Indeed, there are many ways to cover a region with boxes and thus we define N(ε) the minimal number of such boxes required.
The fractal dimension is then defined a s t he l imit a s o ur b ox-size p arameter ε t ends to zero To numerically analyze the fractal dimension of our binary images, we used the FIJI/ImageJ plugin FracLac, developed by A. Karperien.
We shall now try to illustrate this concept by studying a few examples.

Fractal dimension does not depend on mass
Let us consider the following two binary images in Figure 1.To determine fractal dimension, we proceed as explained before.We may identify the domain of each image with the unit square Ω = [0, 1] 2 .
Then, we have to fix ε > 0 and ask ourselves how many squares fit into the white part of each image, respectively.Into the entire domain [0, 1] 2 we could fit approximately 1/ε 2 1 2 DETAILS ON FRACTAL ANALYSIS many squares, since each such square has a volume ε 2 and the volume of the full square is one.
Hence, since roughly 1/4 of the full square is white on the picture on left and 1/2 of the full square on the picture on the right, we find Inserting this into into the definition of the fractal dimension, we find for the image on the left Similarly, for the image on the right Since the answer in either case is 2, we see that the fractal dimension does not depend on the volume of the white mass.In fact, it only tells us that in either case the white mass is a two-dimensional object.

Generating images of arbitrary fractal dimension
A mathematical way of generating images of arbitrary fractal dimension in the interval [1,2] is to use fractal Brownian motion (fBm) first introduced in [25].Fractal Brownian motion is a Gaussian process characterized by a parameter H ∈ (0, 1) called the Hurst parameter.It can be proven that the graph of fractal Brownian motion with Hurst parameter H has fractal dimension 2 − H.The graph of two different fBms is illustrated in Figure 2.

Figure 1 .
Figure 1.Two binary images of same fractal dimension 2, even though the area of the two white areas is different.

Figure 2 .
Figure 2. fBm with Hurst parameter H = 0.7 on the left and with H = 0.3 on the right with same runtime.We emphasize the different scaling of the axes.